#6446720

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,22)(12,18)(13,19)(14,21)(15,20)(16,25)(17,29)(23,38)(24,33)(26,34)(27,31)(28,37)(30,36)(32,39)(35,40);(2,5)(6,7)(9,11,15,24,14,26,13,25)(10,16,19,34,21,33,20,22)(12,27,30,40,29,39,28,23)(17,35,36,31,18,38,37,32);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,23)(13,28)(14,29)(15,30)(16,31)(18,21)(19,36)(20,37)(22,35)(24,27)(25,39)(26,40)(32,33)(34,38);(9,12)(10,18)(11,23)(13,28)(14,29)(15,30)(16,32)(17,21)(19,37)(20,36)(22,38)(24,27)(25,39)(26,40)(31,33)(34,35);(9,13,14,15)(10,19,21,20)(11,24,26,25)(12,28,29,30)(16,22,33,34)(17,36,18,37)(23,27,40,39)(31,35,32,38)

Polycyclic presentation:

2^9|b^2=fg,c^2=gi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353916);